Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields

A simple scheme of unified description of different physical phenomena by using the Langevin type equations is reviewed. Within this approach much attention is being paid to the study of Brownian and quantum motions. Stochastic equations with a white noise term give all characteristics of the Brownian motion. Some generalization of the Langevin type equations allows us to obtain nonlinear equations of particles' motion, which are formally equivalent to the Schrödinger equation. Thus, we establish Nelson's stochastic mechanics on the basis of the Langevin equation.

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Interconnection between Wick multiplication and integration on spaces of nonregular generalized functions in the Lévy white noise analysis

Carpathian Mathematical Publications ◽

10.15330/cmp.11.1.70-88 ◽

2019 ◽

Vol 11 (1) ◽

pp. 70-88

Author(s):

N.A. Kachanovsky ◽

T.O. Kachanovska

Keyword(s):

White Noise ◽

Stochastic Integral ◽

Noise Analysis ◽

Stochastic Equations ◽

Generalized Functions ◽

Wick Product ◽

Pettis Integral ◽

White Noise Analysis ◽

Lévy White Noise

We deal with spaces of nonregular generalized functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to describe a relationship between Wick multiplication and integration on these spaces. More exactly, we show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); and prove a theorem about a representation of the extended stochastic integral via the Pettis integral from the Wick product of the original integrand by a Lévy white noise. As examples of an application of our results, we consider some stochastic equations with Wick type nonlinearities.

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Максимум энтропии в масштабно-инвариантных процессах с 1/f-=SUP=-alpha-=/SUP=--спектром мощности: влияние анизотропии белого шума

Письма в журнал технической физики ◽

10.21883/pjtf.2019.09.47707.17718 ◽

2019 ◽

Vol 45 (9) ◽

pp. 19

Author(s):

В.П. Коверда ◽

В.Н. Скоков

Keyword(s):

Phase Transitions ◽

White Noise ◽

Power Spectra ◽

Random Processes ◽

Stochastic Equations ◽

Alpha Power ◽

Entropy Maximum ◽

The Stability ◽

Nonlinear Stochastic Equations

Large value fluctuations are modeled by a system of nonlinear stochastic equations describing the interacting phase transitions. Under the action of anisotropic white noise, random processes are formed with the 1/f^alpha dependence of the power spectra on frequency at values of the exponent from 0.7 to 1.7. It is shown that fluctuations with 1/f^alpha power spectra in the studied range of changes correspond to the entropy maximum, which indicates the stability of processes with 1/f^alpha power spectra at different values of the exponent alpha.

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Galerkin methods for fractional-stochastic systems

10.22541/au.162289640.05143749/v1 ◽

2021 ◽

Author(s):

Mehmet Ali Akinlar ◽

Francisco Gómez ◽

Fatih Tasci

Keyword(s):

White Noise ◽

Fractional Order ◽

Stochastic Models ◽

Stochastic Systems ◽

Polynomial Chaos ◽

Stochastic Equations ◽

Galerkin Methods ◽

Fractional Order Systems ◽

Fractional Order Derivative ◽

Noise Term

Applicability of undetermined coefficients methods to several fractional-stochastic models is investigated. These models are mostly generated by fractional-order derivative operators and include a fractional white noise term. Application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems are also investigated. Fractional-stochastic equations considered in this paper are totally original systems which may serve as models for many scientific and engineering phenomena. It is pointed out that Galerkin type methods employed in this paper may be efficiently applied to fractional-order systems having uncertainty or a noise term.

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On operators of stochastic differentiation on spaces of regular test and generalized functions of Lévy white noise analysis

Carpathian Mathematical Publications ◽

10.15330/cmp.6.2.212-229 ◽

2014 ◽

Vol 6 (2) ◽

pp. 212-229 ◽

Cited By ~ 1

Author(s):

M.M. Dyriv ◽

N.A. Kachanovsky

Keyword(s):

White Noise ◽

Stochastic Integral ◽

Noise Analysis ◽

Gaussian White Noise ◽

Stochastic Equations ◽

Generalized Functions ◽

White Noise Analysis ◽

Unbounded Operators ◽

Lévy White Noise ◽

Stochastic Derivative

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. In this paper we introduce and study bounded and unbounded operators of stochastic differentiation in the Levy white noise analysis. More exactly, we consider these operators on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using the Lytvynov's generalization of the chaotic representation property. This gives a possibility to extend to the Levy white noise analysis and to deepen the corresponding results of the classical white noise analysis.

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White Noise Calculus in Applications to Stochastic Equations in Hilbert Spaces

Journal of Mathematical Sciences ◽

10.1007/s10958-016-3038-0 ◽

2016 ◽

Vol 218 (4) ◽

pp. 395-429 ◽

Cited By ~ 1

Author(s):

I. V. Melnikova ◽

M. A. Alshanskiy

Keyword(s):

White Noise ◽

Hilbert Spaces ◽

Stochastic Equations ◽

White Noise Calculus

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Attractors for the Stochastic 3D Navier–Stokes Equations

Stochastics and Dynamics ◽

10.1142/s0219493703000772 ◽

2003 ◽

Vol 03 (03) ◽

pp. 279-297 ◽

Cited By ~ 16

Author(s):

Pedro Marín-Rubio ◽

James C. Robinson

Keyword(s):

White Noise ◽

Global Attractor ◽

Weak Solutions ◽

Stokes Equations ◽

Stochastic Equations ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Additive White Noise ◽

Solutions Of Equations

In a 1997 paper, Ball defined a generalised semiflow as a means to consider the solutions of equations without (or not known to possess) the property of uniqueness. In particular he used this to show that the 3D Navier–Stokes equations have a global attractor provided that all weak solutions are continuous from (0, ∞) into L2. In this paper we adapt his framework to treat stochastic equations: we introduce a notion of a stochastic generalised semiflow, and then show a similar result to Ball's concerning the attractor of the stochastic 3D Navier–Stokes equations with additive white noise.

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STOCHASTIC HEAT EQUATION ON ALGEBRA OF GENERALIZED FUNCTIONS

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025712500269 ◽

2012 ◽

Vol 15 (04) ◽

pp. 1250026 ◽

Cited By ~ 9

Author(s):

ABDESSATAR BARHOUMI ◽

HABIB OUERDIANE ◽

HAFEDH RGUIGUI

Keyword(s):

Heat Equation ◽

Diffusion Process ◽

White Noise ◽

Lie Algebra ◽

Stochastic Equations ◽

Generalized Functions ◽

Stochastic Heat Equation ◽

Renormalization Procedure ◽

Poisson Equations

The main objective of this paper is to investigate an extension [Formula: see text] of the "Volterra-Gross" Laplacian on nuclear algebra of generalized functions. In so doing, without using the renormalization procedure, this extension provides a continuous nuclear realization of the square white noise Lie algebra obtained by Accardi–Franz–Skeide in Ref. 2. An extended-Gross diffusion process driven by a class of Itô stochastic equations is studied, and solution of the related Poisson equations is derived in terms of a suitable λ-potential.

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